Lecture 24 External Kink Modes

← Previous Lecture Contents Next Lecture →

Lecture 24
External Kink Modes

Overview

Only after the fixed-boundary problem has been cleared does it make sense to let the plasma edge move. In the external problem \(\xi _a\) is free, the vacuum and wall contribute to \(\delta W\), and instability occurs when the full plasma-plus-vacuum functional becomes negative for a mode with \(\xi _a\neq 0\).

Historical Perspective

The external kink is where Newcomb’s framework becomes practically important. The interior solution alone is not enough; one must also solve the vacuum problem and impose the wall boundary condition. That logic, already clear in cylindrical pinch theory, carried directly into tokamak design: internal profiles determine whether a regular plasma solution exists, while the vacuum region and conducting structures decide whether the boundary can bulge outward. That is one reason this sequence of lectures is so central historically.

24.1 Vacuum energy and the surface form of \(\delta W\)

Vacuum field. Let the plasma occupy \(0\le r\le a\), with vacuum in \(a<r<b\) and a perfectly conducting wall at \(r=b\). In vacuum,

\B _1 = -\grad \Phi , \qquad \lap \Phi = 0. \tag{24.1}

For a single helical harmonic,

\Phi (r,\theta ,z)=\phi (r)e^{i(m\theta +kz)}, \tag{24.2}

\frac {1}{r}\dd {}{r}\left (r\dd {\phi }{r}\right ) - \left (\frac {m^2}{r^2}+k^2\right )\phi =0. \tag{24.3}

The exact finite-\(k\) solutions are modified Bessel functions. For the large-aspect-ratio kink problem, however, \(|ka|\ll 1\), and it is more useful to display the long-wavelength form immediately:

\phi (r)=A r^{|m|}+B r^{-|m|}. \tag{24.4}

Wall condition. At a perfectly conducting wall,

B_{1r}(b)=-\pp {\Phi }{r}\bigg |_{r=b}=0 \qquad \Longrightarrow \qquad \phi '(b)=0. \tag{24.5}

Applying (24.5) to (24.4) gives

B = A\,b^{2|m|},

so that

\phi (r)=A\left (r^{|m|}+b^{2|m|}r^{-|m|}\right ).

Matching at the plasma boundary. At \(r=a\), ideal MHD requires

B_{1r}(a)= iF(a)\,\xi _a. \tag{24.8}

Since \(B_{1r}(a)=-\phi '(a)e^{i(m\theta +kz)}\),

-\phi '(a)=iF(a)\,\xi _a. \tag{24.9}

This fixes the constant \(A\), and evaluating the vacuum magnetic energy then produces a pure boundary term,

\delta W_v = \frac {\pi }{\muo }\,\frac {a^2F^2(a)}{|m|}\,\Lambda _m\,|\xi _a|^2, \tag{24.10}

where

\Lambda _m = \frac {1+\alpha _m}{1-\alpha _m}, \qquad \alpha _m\equiv \left (\frac {a}{b}\right )^{2|m|}. \tag{24.11}

The no-wall limit is \(\Lambda _m\rightarrow 1\).

Full reduced functional. Including the plasma term (23.92), the total energy is

\frac {\delta W}{2\pi ^2 R_0/\muo } = \int _0^a\left (f|\xi '|^2+g|\xi |^2\right )dr + \left [ \left (\frac {FF^\dagger }{k_0^2}\right )_{r=a} + \frac {a^2F^2(a)}{|m|}\Lambda _m \right ]|\xi _a|^2. \tag{24.12}

This is the master expression for the external ideal problem.

24.2 From the internal solution to the surface energy

What is assumed at this stage. At this point the local pressure-driven tests of Lecture 25 have been passed and the fixed-boundary internal problem of Lecture 26 has no unstable mode. One now reopens the variational problem by allowing the edge displacement \(\xi _a\) to be nonzero and by keeping the vacuum term in (24.12).

Convert the bulk term to a boundary term. Starting from

I\equiv \int _0^a\left (f|\xi '|^2+g|\xi |^2\right )dr,

use (23.99) in the form \(g\xi =(f\xi ')'\):

I = \int _0^a \left (f|\xi '|^2+(f\xi ')'\xi ^*\right )dr = \left [f\,\xi '\,\xi ^*\right ]_0^a .

Regularity makes the \(r=0\) term vanish, so

\int _0^a\left (f|\xi '|^2+g|\xi |^2\right )dr = f(a)\,\xi _a'\,\xi _a^*. \tag{24.15}

Substituting (24.15) into (24.12) yields the pure surface form

\frac {\delta W}{2\pi ^2 R_0/\muo } = \left [ \left (\frac {F^2}{k_0^2}\frac {r\xi '}{\xi }\right )_{r=a} + \left (\frac {FF^\dagger }{k_0^2}\right )_{r=a} + \frac {a^2F^2(a)}{|m|}\Lambda _m \right ]|\xi _a|^2. \tag{24.16}

Everything has now been reduced to the boundary behavior of the minimizing interior solution.

Caution

Equation (24.16) is powerful precisely because it hides the labor in the interior problem. The ratio \(a\xi '/\xi \) at the boundary is not arbitrary. It encodes the entire internal profile through the solution of (23.99). Also, the sign of \(\xi _a\) is just a normalization choice: the external question is whether the free-boundary problem admits \(\xi _a\neq 0\) with total \(\delta W<0\).

24.3 External kink and the role of the wall

No-wall \(m=1\) limit. If the plasma is internally stable and \(m=1\), the minimizing interior eigenfunction is rigid to leading order, so \(\xi '=0\) while the edge displacement \(\xi _a\) is now free to be nonzero. Using (24.16) with \(\Lambda _1=1\) gives

\frac {\delta W}{2\pi ^2 R_0/\muo } = \frac {2B_0^2 a^2}{R_0^2} \left (1-\frac {1}{q_a}\right )|\xi _a|^2. \tag{24.17}

Hence

\[q_a>1\]

is the no-wall large-aspect-ratio external-kink threshold. This is exactly the same upper root that appeared in the periodic Kruskal–Shafranov discussion, Eq. (22.39).

A completely explicit analytic profile. To make the full wall problem transparent, take the simplest analytic current profile: constant \(q(r)=q_a\), corresponding in the cylindrical model to a uniform axial current density. Then the factor \(\left (n-\frac {m}{q_a}\right )^2\) is constant, and the Euler–Lagrange equation (23.99) becomes

\dd {}{r}\left (r^3\dd {\xi }{r}\right )-(m^2-1)r\,\xi =0. \tag{24.19}

Try a power law \(\xi \sim r^s\). Then

\[s(s+2)=m^2-1,\]

s=m-1 \qquad \text {or}\qquad s=-m-1.

Regularity at the axis selects

\xi (r)\propto r^{m-1}, \qquad \frac {a\xi '(a)}{\xi (a)}=m-1. \tag{24.22}

Boundary energy for the analytic profile. Insert (24.22) into (24.16). Using (26.5)–(26.6) and \(k_0^2a^2\simeq m^2\) gives

\begin{aligned}\frac {\delta W}{2\pi ^2 R_0/\muo } &= \frac {B_0^2a^2}{R_0^2 m^2 q_a^2} \Big [ (m-1)(nq_a-m)^2 + (n^2q_a^2-m^2) \nonumber \\ &\hspace {3.3cm} + m\Lambda _m (nq_a-m)^2 \Big ]|\xi _a|^2 .\end{aligned} \tag{24.23}

Now factor the bracket carefully:

\begin{aligned}&(m-1)(nq_a-m)^2+(n^2q_a^2-m^2)+m\Lambda _m(nq_a-m)^2 \nonumber \\ &\qquad = m(nq_a-m)\left [(\Lambda _m+1)(nq_a-m)+2\right ].\end{aligned}

Therefore

\frac {\delta W}{2\pi ^2 R_0/\muo } = \frac {B_0^2a^2}{R_0^2 m q_a^2} (nq_a-m)\left [(\Lambda _m+1)(nq_a-m)+2\right ]|\xi _a|^2. \tag{24.25}

Using (24.11),

\Lambda _m+1 = \frac {2}{1-\alpha _m}, \qquad \alpha _m=\left (\frac {a}{b}\right )^{2m},

so (24.25) becomes

\frac {\delta W}{2\pi ^2 R_0/\muo } = \frac {2B_0^2a^2}{R_0^2 m q_a^2(1-\alpha _m)} \, (nq_a-m) \left [nq_a-\left (m-1+\alpha _m\right )\right ] |\xi _a|^2. \tag{24.27}

Hence the ideal external mode is unstable only in the interval

m-1+\alpha _m < nq_a < m, \qquad \alpha _m=\left (\frac {a}{b}\right )^{2m}, \tag{24.28}

or, equivalently,

\boxed { \frac {m-1+\left (a/b\right )^{2m}}{n}<q_a<\frac {m}{n}. } \tag{24.29}

Separate the no-wall and wall pieces. For later resistive-wall use, it is convenient to write the ideal-wall result as

\delta W_{\rm ideal}(b)=\delta W_\infty ^{\rm tok}+\delta W_b^{\rm tok}. \tag{24.30}

Setting \(\alpha _m=0\) in (24.27) gives the no-wall energy

\frac {\delta W_\infty ^{\rm tok}}{2\pi ^2 R_0/\muo } = \frac {2B_0^2a^2}{R_0^2 m q_a^2} \, (nq_a-m)\left [nq_a-(m-1)\right ] |\xi _a|^2, \tag{24.31}

while the wall contribution is the difference

\frac {\delta W_b^{\rm tok}}{2\pi ^2 R_0/\muo } = \frac {2B_0^2a^2}{R_0^2 m q_a^2} \, \frac {\alpha _m}{1-\alpha _m}(nq_a-m)^2 |\xi _a|^2. \tag{24.32}

The resistive-wall regime is the strip

m-1<nq_a<m-1+\alpha _m, \tag{24.33}

where \(\delta W_\infty ^{\rm tok}<0\) but \(\delta W_{\rm ideal}(b)=\delta W_\infty ^{\rm tok}+\delta W_b^{\rm tok}>0\).

Connection back to Kruskal–Shafranov. For \(m=n=1\), Eq. (24.29) reduces to

\left (\frac {a}{b}\right )^2<q_a<1,

which is exactly the periodic ideal-wall stability band derived earlier in Eq. (22.62). For general \((m,n)\) the same algebra produces a band immediately below each rational value \(q_a=m/n\). This is the diffuse-profile version of the same story: the wall introduces a second root, and that second root is what creates the stability window.

Interactive External-Kink Explorer

Open a browser companion to the external-kink lecture. The app reproduces the lecture’s ideal-wall constant-\(q\) bands in \(q_a\), then replaces the analytic interior with a smooth tokamak current profile fixed by \(q_0\) and \(q_a\) so the edge logarithmic derivative \(a\xi\prime(a)/\xi(a)\) comes from a numerical Newcomb solve.

Open the external-kink explorer

Figure 24.1: Comparison of external-kink growth-rate bands for analytic profile models with \(a/b=0.9\). Panel (a) shows the tokamak-like constant-\(q\) case with \(n=1\) and several poloidal mode numbers \(m\), using Panel (b) shows the RFP-like \(m=1\) branch for several toroidal mode numbers \(n\), using the rigid-core summary form of the constant-\(\lambda \) analogue, For reversed field, \(q_a<0\), the unstable branches correspond to negative \(n\), so the toroidal mode number changes sign together with the edge safety factor. In both panels, growth occurs only where the plotted expression is positive.

Kinetic energy and growth rate. For a normal mode with time dependence \(e^{\gamma t}\), the ideal-MHD energy principle gives

\delta W + \gamma ^2 \mathcal {K} = 0, \qquad \mathcal {K} \equiv \frac 12 \int \rho \, |\bm {\xi }|^2\, dV .

For the analytic constant-\(q\) profile, the regular internal solution is

\xi _r(r) = \xi _a \left (\frac {r}{a}\right )^{m-1}.

If we approximate the inertia by the dominant radial displacement and take uniform density, then

\begin{aligned}\mathcal {K} &= \frac 12 \int _0^{2\pi } d\theta \int _0^{2\pi R_0} dz \int _0^a \rho \, |\xi _r|^2\, r\,dr \nonumber \\ &= 2\pi ^2 R_0 \rho \int _0^a |\xi _a|^2 \left (\frac {r}{a}\right )^{2m-2} r\,dr \nonumber \\ &= \frac {\pi ^2 \rho R_0 a^2}{m}\, |\xi _a|^2.\end{aligned}

Using Eq. (24.27),

\delta W = \frac {4\pi ^2 B_0^2 a^2}{\mu _0 R_0\, m q_a^2(1-\alpha _m)} \, (nq_a-m)\left [nq_a-(m-1+\alpha _m)\right ] |\xi _a|^2, \qquad \alpha _m=\left (\frac {a}{b}\right )^{2m}.

Therefore

\gamma ^2 = -\frac {\delta W}{\mathcal {K}} = -\frac {4 B_0^2}{\mu _0 \rho R_0^2 q_a^2(1-\alpha _m)} \, (nq_a-m)\left [nq_a-(m-1+\alpha _m)\right ].

Writing \(v_A^2 = B_0^2/(\mu _0 \rho )\), this becomes

\boxed { \left (\frac {\gamma R_0}{v_A}\right )^2 = \frac {-4 (nq_a-m)\left [nq_a-(m-1+\alpha _m)\right ]} {q_a^2(1-\alpha _m)} } \qquad \left (\alpha _m=\left (\frac {a}{b}\right )^{2m}\right ).

Hence instability requires \(\gamma ^2>0\), i.e.

\frac {m-1+\alpha _m}{n} < q_a < \frac {m}{n}.

The resistive-wall continuation. Once the shell conductivity is finite, the ideal-wall increment \(\delta W_b^{\rm tok}\) is only partially restored. The thin-wall interpolation and dispersion relation derived in the Kruskal–Shafranov lecture, Eqs. (22.107) and (22.110), therefore give

\gamma \tau _w = -\frac {2b^2}{b^2-a^2} \frac {\delta W_\infty ^{\rm tok}}{\delta W_{\rm ideal}(b)}. \tag{24.42}

That is the slow RWM branch sitting just below the ideal-wall threshold. The same bookkeeping will reappear in Appendix E, where Eqs. (E.44) and (E.45) give the corresponding \(m=1\) RFP energies.

Constant-\(\lambda \) RFP analogue. The force-free Taylor equilibrium \(\curl \B =\lambda \B \) gives the RFP companion to the analytic constant-\(q\) exercise. Appendix E works the derivation through explicitly, starting from the Bessel core (E.26) and the edge safety-factor relation (E.32) and ending at Eqs. (E.37)–(E.42). In the same rigid-core \(m=1\) approximation used here, the result can be written as

\frac {\delta W_{m=1}^{\rm RFP}}{2\pi ^2 R_0/\muo } \simeq \frac {2B_{\theta a}^2}{1-\alpha _1} \,(nq_a-1)(nq_a-\alpha _1)\,|\xi _a|^2, \qquad \alpha _1=\left (\frac {a}{b}\right )^2, \tag{24.43}

with \(B_{\theta a}\equiv B_\theta (a)\). For a uniform-density rigid core,

\left (\frac {\gamma a}{v_{A\theta a}}\right )^2 \simeq \frac {-4 (nq_a-1)(nq_a-\alpha _1)}{1-\alpha _1}, \qquad v_{A\theta a}^2\equiv \frac {B_{\theta a}^2}{\mu _0\rho }. \tag{24.44}

So the same surface-energy logic predicts a low-\(q\) RFP band \(\alpha _1<nq_a<1\): the qualitative difference is that \(q_a<1\) lets several toroidal harmonics fit below the same upper root.

What these analytic profiles are good for. The constant-\(q\) profile is not the only choice. A constant-\(\lambda \) force-free profile leads to the familiar Bessel-function equilibrium, and its RFP free-boundary reduction is worked through in Appendix E, Eqs. (E.28)–(E.50). For the variational logic in the tokamak setting, the constant-\(q\) example is hard to beat because it makes the boundary ratio \(a\xi '/\xi \) analytic and therefore makes the stability bands, the no-wall/wall split, and the RWM continuation completely explicit.

Takeaways

This four-lecture sequence is the real payoff of Newcomb’s framework. We find:
1.
a one-dimensional variational formulation for the screw pinch;
2.
local pressure-driven tests (Suydam and Mercier);
3.
the fixed-boundary \(m=1\) internal kink when \(q_0<1\);
4.
external kink bands below each rational \(q_a=m/n\) once the wall solution is kept, together with the slow resistive-wall continuation when only a fraction of the ideal-wall increment survives.

The wall does not merely shift a number. It changes the factorization of \(\delta W\) and introduces a second root. That is exactly why the diffuse-pinch problem is richer than the surface-current Kruskal–Shafranov model while still remaining close enough to it to teach from.

Bibliography

Problems

Problem 24.1.: Starting from (23.93), rederive (23.99) and (23.100) without skipping the integration-by-parts step.
Problem 24.2.: Starting from (25.8), derive the Suydam criterion (25.17) and verify the factor of \(8\) in front of \(\muo p'/B_z^2\).
Problem 24.3.: Starting from (25.37), show that the toroidal correction relative to Suydam is \(\Delta _{\rm tor}=-q^2p'\). For \(p'<0\), determine the sign of the pressure term for \(q<1\), \(q=1\), and \(q>1\).
Problem 24.4.: For the analytic profile \(q(r)=q_a\), derive (24.27) explicitly for \(m=2\), \(n=1\) and identify the wall-stabilized interval in \(q_a\).
Problem 24.5.: Repeat the vacuum calculation leading to (24.11) for a no-wall plasma by taking \(b\rightarrow \infty \), and show directly that \(\Lambda _m\rightarrow 1\).